This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A276862 #63 Feb 14 2024 17:27:32
%S A276862 2,3,2,3,2,2,3,2,3,2,2,3,2,3,2,3,2,2,3,2,3,2,2,3,2,3,2,3,2,2,3,2,3,2,
%T A276862 2,3,2,3,2,2,3,2,3,2,3,2,2,3,2,3,2,2,3,2,3,2,3,2,2,3,2,3,2,2,3,2,3,2,
%U A276862 2,3,2,3,2,3,2,2,3,2,3,2,2,3,2,3,2
%N A276862 First differences of the Beatty sequence A003151 for 1 + sqrt(2).
%C A276862 Conjectures: Equals both A245219 and A097509. - _Michel Dekking_, Feb 18 2020
%C A276862 Theorem: If the initial term of A097509 is omitted, it is identical to the present sequence. For proof, see A097509. The argument may also imply that A082844 is also the same as these two sequences, apart from the initial terms. - Manjul Bhargava, Kiran Kedlaya, and Lenny Ng, Mar 02 2021. Postscript from the same authors, Sep 09 2021: We have proved that the present sequence, A276862 (indexed from 1) matches the characterization of {c_{i-1}} given by (8) of our "Solutions" page.
%H A276862 Clark Kimberling, <a href="/A276862/b276862.txt">Table of n, a(n) for n = 1..9999</a> [Offset adapted by _Georg Fischer_, Mar 07 2020]
%H A276862 Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, <a href="https://arxiv.org/abs/2402.08331">Beatty Sequences for a Quadratic Irrational: Decidability and Applications</a>, arXiv:2402.08331 [math.NT], 2024. See pp. 17-18.
%H A276862 N. J. A. Sloane, <a href="/A115004/a115004.txt">Families of Essentially Identical Sequences</a>, Mar 24 2021 (Includes this sequence)
%F A276862 a(n) = floor((n+1)*r) - floor(n*r) = A003151(n+1)-A003151(n), where r = 1 + sqrt(2), n >= 1.
%F A276862 a(n) = 1 + A006337(n) for n >+ 1. - _R. J. Mathar_, Sep 30 2016
%F A276862 Fixed point of the morphism 2 -> 2,3; 3 -> 2,3,2. - _John Keith_, Apr 21 2021
%t A276862 z = 500; r = 1+Sqrt[2]; b = Table[Floor[k*r], {k, 0, z}]; (* A003151 *)
%t A276862 Differences[b] (* A276862 *)
%t A276862 Last@SubstitutionSystem[{2 -> {2, 3}, 3 -> {2, 3, 2}}, {2}, 5] (* _John Keith_, Apr 21 2021 *)
%o A276862 (PARI) vector(100, n, floor((n+1)*(1 + sqrt(2))) - floor(n*(1+sqrt(2)))) \\ _G. C. Greubel_, Aug 16 2018
%o A276862 (Magma) [Floor((n+1)*(1 + Sqrt(2))) - Floor(n*(1+Sqrt(2))): n in [1..100]]; // _G. C. Greubel_, Aug 16 2018
%o A276862 (Python)
%o A276862 from math import isqrt
%o A276862 def A276862(n): return 1-isqrt(m:=n*n<<1)+isqrt(m+(n<<2)+2) # _Chai Wah Wu_, Aug 03 2022
%Y A276862 Cf. A003151, A006337, A014176, A082844, A097509, A245219, A276879.
%Y A276862 The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A003151 as the parent: A003151, A001951, A001952, A003152, A006337, A080763, A082844 (conjectured), A097509, A159684, A188037, A245219 (conjectured), A276862. - _N. J. A. Sloane_, Mar 09 2021
%K A276862 nonn,easy
%O A276862 1,1
%A A276862 _Clark Kimberling_, Sep 24 2016
%E A276862 Corrected by _Michel Dekking_, Feb 18 2020